No, C4 is not the smoothing cap. It is rated at 1000pF. C4 should connect to the backplate. On the ELA M 251 schematic, C8 is the cap I'm refering to. I doubt your mic has the smoothing cap. This capacitor is one of the ways AKG (the designers of the 251) was trying to reach their design goals. They were trying to build a competitor for the U-47, for which Telefunken had lost American distribution. The cap shaved off the high end to make it more tonally akin to the U-47 with the much brighter C12 capsule.
In your case, the cap could be on a switch for the brighter horns that come your way. Nice mic, though. If it sounds great to you, the switch might be more trouble than it is worth, but trying out the cap would be easy enough.